1. Ta có: $\int\limits_0^{\frac{\pi}{2}}(x+\cos^3x)\sin xdx$
$=\int\limits_0^{\frac{\pi}{2}}\cos^3x\sin xdx+\int\limits_0^{\frac{\pi}{2}}x\sin xdx$
$=-\int\limits_0^{\frac{\pi}{2}}\cos^3xd(\cos x)-\int\limits_0^{\frac{\pi}{2}}xd(\cos x)$
$=-\dfrac{\cos^4x}{4}\left|\begin{array}{l}\dfrac{\pi}{2}\\0\end{array}\right.-x\cos x\left|\begin{array}{l}\dfrac{\pi}{2}\\0\end{array}\right.+\int\limits_0^{\frac{\pi}{2}}\cos xdx$
$=\dfrac{1}{4}+\sin x \left|\begin{array}{l}\dfrac{\pi}{2}\\0\end{array}\right.$
$=\dfrac{5}{4}$